Reconstruction approximating method by biquadratic splines of offset surfaces holes

Standard Offset surfaces are defined as locus of the points which are at constant distance along the unit normal direction from the generator surfaces. Offset are widely used in various practical applications, such as tolerance analysis, geometric optics and robot path-planning. In some of the engineering applications, we need to extend the concept of standard offset to the generalized offset where distance offset is not necessarily constant and offset direction are not necessarily along the normal direction. Normally, a generalized offset is functionally more complex than its progenitor because of the square root appears in the expression of the unit normal vector. For this, an approximation method of its construction is necessary. In many situation it is necessary to fill or reconstruct certain function defined in a domain in which there is a lack of information inside one or several sub-domains (holes). In some practical cases, we may have some specific geometrical constrains, of industrial or design type, for example, the case of a specified volume inside each one of these holes. The problem of filling holes or completing a 3D surface arises in all sorts of computational graphics areas, like CAGD, CAD-CAM, Earth Sciences, computer vision in robotics, image reconstruction from satellite and radar information, etc. In this work we present an approximation method of filling holes of the generalized offset of a surface when there is a lack information in a sub-domain of the function that define it. We prove the existence and uniqueness of solution of this problem, we show how to compute it and we establish a convergence result of this approximation method. Finally, we give some graphical and numerical examples.

Monte Carlo construction of cubature on Wiener space

In this paper, we investigate application of mathematical optimization to construction of a cubature formula on Wiener space, which is a weak approximation method of stochastic differential equations introduced by Lyons and Victoir (Proc R Soc Lond A 460:169–198, 2004). After giving a brief review on the cubature theory on Wiener space, we show that a cubature formula of general dimension and degree can be obtained through a Monte Carlo sampling and linear programming. This paper also includes an extension of stochastic Tchakaloff’s theorem, which technically yields the proof of our primary result.

The 3-D Image of The Temperature Integral And Its Self-Similarity

How to derive the accurate value of temperature integral is a vital problem for the non-isothermal kinetic analysis. In the past six decades, researchers provided various methods to solve above problem, but the error usually becomes divergent when the value of x (x=Ea/RT) is too small or too large, no matter whether it is a numerical method or an approximation method. In this paper, we present a new series method and elaborately design a computer program to calculate the value of temperature integral. Finally, we reveal the mysterious relationship between the integral, the temperature and the activation energy, and we find an extremely interesting phenomenon that the 3-D image of the temperature integral is of self-similarity according to the fractal theory.

Rational Approximation Method for Stiff Initial Value Problems

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order.

Rational Approximations of Arbitrary Order: A Survey

This paper deals with the study and analysis of several rational approximations to approach the behavior of arbitrary-order differentiators and integrators in the frequency domain. From the Riemann–Liouville, Grünwald–Letnikov and Caputo basic definitions of arbitrary-order calculus until the reviewed approximation methods, each of them is coded in a Maple 18 environment and their behaviors are compared. For each approximation method, an application example is explained in detail. The advantages and disadvantages of each approximation method are discussed. Afterwards, two model order reduction methods are applied to each rational approximation and assist a posteriori during the synthesis process using analog electronic design or reconfigurable hardware. Examples for each reduction method are discussed, showing the drawbacks and benefits. To wrap up, this survey is very useful for beginners to get started quickly and learn arbitrary-order calculus and then to select and tune the best approximation method for a specific application in the frequency domain. Once the approximation method is selected and the rational transfer function is generated, the order can be reduced by applying a model order reduction method, with the target of facilitating the electronic synthesis.